Exact division means remainder zero

3.OA.C.73.OA.B.6 · take · grade 3

Archetype: Divisibility and Remainder Reasoning · step in a 8-type progression

The division below comes out exactly (no remainder). Find every digit that $\blacksquare$ could be. (Here $7\blacksquare$ is a two-digit number.)

$7\blacksquare \div 4$

Show solution

Understand

The two-digit number 7-followed-by-a-blank must divide exactly by 4. We need every digit the blank could be so that the number is a multiple of 4.

Givens

The number is two digits and starts with 7 (it is 70 through 79).
When divided by 4 it must leave no remainder.

Unknowns

All possible digits for the blank (the ones digit).

Constraints

The blank is a single digit 0 through 9.
The whole number must be divisible by 4.

Plan

#2 Make a Systematic List · also uses: #6 Guess and Check

There are only ten numbers from 70 to 79, so we can list the multiples of 4 in that range and read off which ones-digits work.

Execute

#2 Make a Systematic List 3.OA.C.7

The multiples of 4 are 4 times 18 equals 72 and 4 times 19 equals 76. The next one, 4 times 20, is 80, which is past the 70s.

4 \times 18 = 72,\quad 4 \times 19 = 76

Listing the multiples of 4 inside 70 to 79 catches every number that divides evenly.

#6 Guess and Check 3.OA.B.6

Both 72 and 76 start with 7, so the blank can be 2 or 6. Checking 70, 74, 78 shows they leave remainders, so they do not work.

72 \div 4 = 18,\quad 76 \div 4 = 19

Each valid number gives a whole quotient, confirming the digit makes a clean division.

Answer: 2 or 6

Review

72 and 76 are the only multiples of 4 between 70 and 79 (multiples of 4 are spaced 4 apart, so a ten-wide window holds two or three of them). Both give whole quotients, so the answers are correct.

Use the divisibility shortcut for 4: a number is divisible by 4 when its last two digits are. Testing 70..79 by adding 4 from 72 gives 72 and 76 again.

Standards · min grade 3

3.OA.C.7 Fluently multiply and divide within 100 — Listing multiples of 4 (72 and 76) within the 70s.
3.OA.B.6 Understand division as an unknown-factor problem — Checking that each candidate divides by 4 with a whole-number quotient.

💡 This only needs Grade 3 multiplication facts: list the multiples of 4 and see which land in the 70s!